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G-spectrum
In algebraic topology, a G-spectrum is a spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ... with an action of a (finite) group. Let ''X'' be a spectrum with an action of a finite group ''G''. The important notion is that of the homotopy fixed point set X^. There is always :X^G \to X^, a map from the fixed point spectrum to a homotopy fixed point spectrum (because, by definition, X^ is the mapping spectrum F(BG_+, X)^G). Example: \mathbb/2 acts on the complex ''K''-theory ''KU'' by taking the conjugate bundle of a complex vector bundle. Then KU^ = KO, the real ''K''-theory. The cofiber of X_ \to X^ is called the Tate spectrum of ''X''. ''G''-Galois extension in the sense of Rognes This notion is due to J. Rognes . Let ''A'' be an E∞-ring with an action of a finit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrum (topology)
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory\mathcal^*:\text^ \to \text,there exist spaces E^k such that evaluating the cohomology theory in degree k on a space X is equivalent to computing the homotopy classes of maps to the space E^k, that is\mathcal^k(X) \cong \left , E^k\right/math>.Note there are several different categories of spectra leading to many technical difficulties, but they all determine the same homotopy category, known as the stable homotopy category. This is one of the key points for introducing spectra because they form a natural home for stable homotopy theory. The definition of a spectrum There are many variations of the definition: in general, a ''spectrum'' is any sequence X_n of pointed topological spaces or pointed simplicial sets together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |